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Journal of Computational Neuroscience 9, 259–270, 2000
c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.
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Nonlinear Thermodynamic Models of Voltage-Dependent Currents
ALAIN DESTEXHE
Department of Physiology, Laval University, Quebec, Canada G1K 7P4 and Unité de Neurosciences Intégratives
et Computationnelles, CNRS, UPR-2191, Avenue de la Terrasse, 91198 Gif-sur-Yuette, France
[email protected]
JOHN R. HUGUENARD
Department of Neurology and Neurological Sciences, Stanford University, Stanford, CA 94305, USA
Received November 10, 1999; Revised February 17, 2000; Accepted
Action Editor:
Abstract. Hodgkin and Huxley provided the first quantitative description of voltage-dependent currents and
adjusted their model to experimental data using empirical functions of voltage. A physically plausible formalism
was proposed later by assuming that transition rates depend exponentially on a free-energy barrier, by analogy with
the theory of reaction rates. It was also assumed that the free energy depends linearly on voltage. This thermodynamic
formalism can accurately describe many processes, but the resulting time constants can be arbitrarily fast, which
may also lead to aberrant behavior. We considered here a physically plausible solution to this problem by including
nonlinear effects of the electrical field on the free energy. We show that including effects such as mechanical
constraints, inherent to the structure of the ion channel protein, leads to more accurate thermodynamic models.
These models can account for voltage-dependent transitions that are rate-limited in a given voltage range, without
invoking additional states. We illustrate their applicability to fit experimental data by considering the case of the
T-type calcium current in thalamic neurons.
Keywords:
1.
ion channels, gating, Hodgkin-Huxley models, Markov models, T-type calcium currents
Introduction
Hodgkin and Huxley (1952) provided the first quantitative characterization of the voltage dependence of
ionic currents and its role in generating action potentials. The formalism they used invoked the assembly
of several gating particles acting independently in a
voltage-dependent manner. The voltage dependence of
the rate constants were fit to voltage-clamp measurements using empirical functions of voltage, which conferred to the model a behavior consistent with action
potential genesis in squid giant axon.
Instead of using empirical functions, it is also possible to deduce the functional form of the voltage dependence of the rate constants from thermodynamics.1
These thermodynamic models (Eyring et al., 1949;
Tsien and Noble, 1969; Hill and Chen, 1972; Stevens,
1978; Hille, 1992) provide a firm physical basis to constrain and parameterize the voltage dependence of rate
constants, which are then used to fit voltage-clamp data.
Thermodynamic models usually assume that a freeenergy barrier is associated to the transition, by analogy with the theory of reaction rates (Eyring, 1935;
Johnson et al., 1974). They also assume that the effect
of the electrical field is linear on the free energy, which
can be interpreted physically as reducing the conformational change to the translation of a freely moving electric charge or the rotation of rigid dipoles (Andersen
and Koeppe, 1992; Hille, 1992; Johnston and Wu,
1995).
260
Destexhe and Huguenard
Although this approach has proved to be accurate
for many purposes (reviewed in Hille, 1992), it has the
major complication that the time constant can reach
arbitrarily small values, which may lead to aberrant
behavior. In reality, the complex molecular structure
of ion channels imposes bounds on their transition
rates. Because exponential functions of voltage cannot account for rate-limited transitions, this problem
is usually solved by introducing a minimal rate (BorgGraham, 1991), forcing the rate constants to saturate
(Willms et al., 1999), or using additional voltageindependent rate-limiting transitions. These procedures are, however, largely empirical.
In this article, we consider a physically plausible
framework to solve this problem. We introduce a functional form for the voltage dependence of rate constants that takes into account more realistic effect of
the electrical field on the ion channel protein. We then
show that this functional form naturally accounts for
rate-limiting effects, and we illustrate its applicability
by modeling the voltage-dependence of T-type calcium
currents.
2.
Methods
Whole-cell voltage-clamp recordings of the T-type calcium current were obtained from thalamic relay neurons acutely dissociated from the ventrobasal thalamus
of young rats (P8–P15). All voltage-clamp recordings
were at a temperature of 24 degree centigrade. The
methods were described in detail in Huguenard and
Prince (1992).
The voltage-clamp behavior of the T-type Ca2+ current (IT ) was modeled using a Hodgkin and Huxley
(1952) type model. Due to the nonlinear behavior of
Ca2+ currents, IT was represented using the constantfield equations (see Hille, 1992):
Z 2 F 2 V Cai − Cao exp(−ZFV/RT)
,
IT = P̄Ca m h
RT
1 − exp(−ZFV/RT)
(1)
2
where P̄Ca (in centimeter per second) is the maximum
permeability of the membrane to Ca2+ ions, Z = 2 is
the valence of Ca2+ ions, F is the Faraday constant,
R is the gas constant, and T is the absolute temperature in degrees Kelvin. Cai and Cao are the intracellular and extracellular Ca2+ concentrations (in M),
respectively. The gating of IT was represented by two
inactivation gates (m) and one inactivation gate (h). m
and h are the fraction of each gate in the open configuration. The m 2 h formalism in Eq. (1) was deduced
by maximum likelihood criteria from voltage-clamp
experiments (Huguenard and Prince, 1992). These
variables were described by the following first-order
equations:
dm
= αm (V )(1 − m) − βm (V )m
dt
dh
= αh (V )(1 − h) − βh (V )h,
dt
(2)
(3)
where αm and βm are, respectively, the forward and
backward rate constants for activation. For the inactivation gate, these constants are αh and βh , respectively.
The quantities directly observable using voltageclamp experiments are the steady-state activation
(m ∞ ), the activation time constant (τm ), the steadystate inactivation (h ∞ ), and the inactivation time constant (τh ). These quantities are given, respectively, by
m ∞ = αm /[αm + βm ]
(4)
τm = 1/[αm + βm ]
(5)
h ∞ = αh /[αh + βh ]
(6)
τh = 1/[αh + βh ].
(7)
These equations were applied to model the T-type
calcium current in thalamic neurons (see Section 3,
Results), using different methods to determine the exact functional form of the rate constants αm , βm , αh ,
and βh . One of these methods consisted of an empirical fit to the experimental data (Huguenard and
McCormick, 1992). The steady-state relations were
fit using Boltzmann functions (Fig. 2A-B, solid lines),
leading to the following optimal functions:
m ∞ (V ) = 1/(1 + exp[−(V + 57)/6.2])
h ∞ (V ) = 1/(1 + exp[(V + 81)/4]).
The voltage-dependence of time constants was represented by multiexponential functions that were fit to
the experimental data. (Huguenard and McCormick,
1992), leading to the following expression for activation:
τm (V ) = 0.612 + 1/(exp[−V + 312)/16.7]
+ exp[(V + 16.8)/18.2])
(8)
Nonlinear Thermodynamic Models of Ionic Currents
3.1.
and for inactivation:
τh (V ) = 28 + exp[−(V + 22)/10.5]
for V ≥ −81mV
exp[(V + 467)/66.6]
(9)
for V < −81mV.
Here, two different functions were fit to the time constants τh obtained from inactivation protocols (V ≥ −81
mV) or recovery from inactivation (V < −81 mV).
The T-current was simulated in current-clamp in a
single-compartment neuron described by the following
equation:
Cm
dV
= −g L (V − E L ) − IT ,
dt
(10)
where Cm = 0.88 µF/cm2 is the membrane capacitance, g L = 0.038 mS/cm2 and E L = −77 mV are
the leak conductance and reversal potential, and IT is
the T-current as given by Eq. (1). These parameters
were obtained by matching the model to thalamic neurons recorded in vitro (Destexhe et al., 1998).
All models were simulated using the NEURON simulation environment, which can solve cable equations
and voltage-dependent conductances based on either
differential equations or kinetic diagrams (Hines and
Carnevale, 1997). Models were fit to experimental data
using a simplex search procedure (Press et al., 1986).
Fitting was performed simultaneously on two data sets:
time constants and steady-state values (see Fig. 2).
Each data set was given equal weight by renormalizing
its mean-square error by its maximal amplitude. All
computational models and fitting procedures were run
on a Sparc 20 workstation (Sun Microsystems, Mountain View, CA).
3.
Results
We first describe the empirical fitting of a HodgkinHuxley type model to the T-type calcium current in
thalamic neurons. We then consider linear and nonlinear thermodynamic models and compare their adequacy to model the voltage-dependent properties of the
T-current.
261
Hodgkin-Huxley Model of the T-Type
Calcium Current
The T-type Ca2+ calcium current (also called lowthreshold Ca2+ current) is responsible for the genesis
of bursts of action potentials in many cell types, such as
thalamic neurons (Jahnsen and Llinás, 1984). In thalamic neurons, this current was characterized by a number of voltage-clamp studies (reviewed in Huguenard,
1996). Several models of the T-current were proposed
(reviewed in Destexhe and Sejnowski, 1997). To represent the “classical” Hodgkin-Huxley approach, which
consists of adjusting rate constants to experimental data
by using empirical functions of voltage (Hodgkin and
Huxley, 1952), we have used here a model drawn by
Huguenard and McCormick (1992), which is given in
Section 2, Methods.
The T-type Ca2+ current is transient and has activation and inactivation characteristics similar to the
fast Na+ current but is slower, and its voltage range
for activation and inactivation typically occurs around
rest. Voltage-clamp recordings of the T-current are
shown in Fig. 1A. A typical activation protocol for the
T-current, consisting of a series of voltage steps from
a hyperpolarized level (−100 mV) to various depolarized levels, reveals an inward current that activates and
inactivates in a voltage-dependent manner (Fig. 1A1).
A particular feature of the T-current is that the recovery
from inactivation is very slow. This is shown using a
voltage-clamp protocol in which the current is reactivated from a holding voltage where the current is fully
inactivated (Fig. 1A2). The time constant of recovery,
estimated from fitting an exponential to the peak currents (dashed line in Fig. 1A2), reveals time constants
of about 300 ms, almost an order of magnitude slower
than inactivation time constants.
The voltage-dependent properties of this current
are illustrated in Fig. 2 (symbols). The steady-state
relations and time constants obtained from voltageclamp recordings of IT are shown for several cells.
Steady-state activation and inactivation had a sigmoidal
voltage-dependence (Fig. 2A-B, symbols) consistent
with the characteristics of the T-current in other types of
neurons (Huguenard, 1996). Activation time constants
showed a bell-shaped curve with values ranging from
1.5 to 20 ms and reached a constant value around 1.5 ms
at depolarized levels (Fig. 2C, symbols). Inactivation
time constants ranged from 25 ms to 120 ms (symbols
> −81 mV in Fig. 2D), also showing a saturation to a
262
Destexhe and Huguenard
Figure 1. Experiments and models of the voltage-clamp behavior of the T-current in thalamic neurons. Left panels: Activation protocol.
Command potentials at various levels were given after the cell was maintained at a hyperpolarized holding potential, leading to the activation
of the current. Right panels: Protocol for the recovery from inactivation. From a holding voltage of −40 mV at which the current is fully
inactivated, the current is stepped to −90 mV for variable durations, then stepped back to −40 mV. A: Experimental data from isolated thalamic
neurons at 24◦ C. B: Empirical Hodgkin-Huxley model (see Section 2, Methods; the density of T-channels in B was P̄Ca = 3 × 10−6 cm/s).
constant value (about 25 ms) at depolarized levels and
a slow recovery (symbols ≤ −81 mV in Fig. 2D).
Thus the T-current in thalamic relay neurons has activation and inactivation that are characterized by relatively slow time constants and a slow recovery from
inactivation, almost an order of magnitude slower than
inactivation.
The fit of a Hodgkin-Huxley model to these experimental data using empirical functions of voltage
(see Section 2, Methods) is shown in Fig. 2 (solid
lines). The behavior of this model in voltage clamp
(Fig. 1B) accounted well for all voltage-clamp protocols with activation and recovery from inactivation
shown in Fig. 1B1 and B2, respectively. However,
in this model, τm and τh were fit using functions of
voltage obtained empirically. Similar to the work of
Hodgkin and Huxley (1952), this approach leads to a
model that accounts well for the current-clamp behavior of the T-current in thalamic neurons (McCormick
and Huguenard, 1992; see below).
3.2.
Thermodynamic Models
Another possibility is to deduce the exact functional
form of the voltage-dependence of the model from
thermodynamics. This type of model assumes that the
gating of ion channels operates through successive conformational changes of the ion channel protein. Consider an elementary transition between an initial (I )
and a final (F) state, with a rate constant r (V ) that is
voltage-dependent:
r(V)
I −→ F.
(11)
According to the theory of reaction rates (Eyring, 1935;
Johnson et al., 1974), the rate of the transition depends
exponentially on the free energy barrier between the
two states:
r (V ) = r0 e−1G(V )/RT ,
(12)
Nonlinear Thermodynamic Models of Ionic Currents
263
Figure 2. Fitting of differnt models to the T-current in thalamic relay neurons. In each panel, the symbols show the voltage-clamp data
obtained in several thalamic neurons (from Huguenard and McCormick, 1992), and the continuous curves show the best fits obtained with
three types of Hodgkin-Huxley models: an empirical model (solid lines), a linear thermodynamic model (thin dashed lines) and a nonlinear
thermodynamic model (thick dashed lines). A: Steady-state activation (m 2∞ ). B: Steady-state inactivation (h ∞ ). C: Activation time constant
(τm ). D: Inactivation time constant (τh ). The leftmost symbols in D (≤−81 mV) are the data from the slow recovery from inactivation of the
T-current. See text for the values of the parameters.
264
Destexhe and Huguenard
Stevens, 1978; Andersen and Koeppe, 1992). For example, linear terms in V will result if the conformations
differ in their net number of charges or if the conformational change is accompanied with the translation
of a freely moving charge inside the structure of the
channel (see Fig. 5A).
Applying Eqs. (12) to (14), the rate constant can be
written as
r (V ) = r0 e−[(A
∗
+B ∗ V )−(A0 +B0 V )]/RT
= r0 e−(a+bV )/RT ,
Figure 3. Schematic representation of the free energy profile of
conformational changes in ion channels. The diagram represents the
free energy of different states involved in a transition: the initial
state, activated complex, and final state. The equilibrium distribution
between initial and final states depends on the relative value of their
free energy (G 0 and G 1 ). The rate of the transition will be governed
by the free energy barrier 1G, which is the free energy difference
between the activated complex and the initial state.
where r0 is a constant and 1G(V ) is the free energy
barrier, which can be written as
1G(V ) = G ∗ (V ) − G 0 (V ),
(13)
where G ∗ (V ) is the free energy of an intermediate state
(activated complex) and G 0 (V ) is the free energy of
the initial state, as illustrated in Fig. 3. The relative
values of the free energy of the initial and final states
(G 0 and G 1 ) determine the equilibrium distribution between these states, while the kinetics of the transition
depend on the size of the free-energy barrier 1G(V ).
Linear Thermodynamic Models It is usually assumed
that a linear voltage dependence of the free energy is
sufficient to describe the gating of ion channels (Tsien
and Noble, 1969; Stevens, 1978; Borg-Graham, 1991;
Andersen and Koeppe, 1992; Hille, 1992). According
to these linear thermodynamic models, the free energy
of a given state i can be written as
G i (V ) = Ai + Bi V,
,
(15)
where a = A∗ − A0 and b = B ∗ − B0 represent differences between the constant and linear components
of the free energy of the initial and activated states
(according to Eq. (14)).
Consider the particular case of a reversible openclosed transition
α(V )
C −→
←− O,
(16)
β(V )
where C and O are, respectively, the closed and open
states, and α and β are the forward and backward
rate constants. Applying Eq. (15) to forward and backward reactions leads to the following expression for the
voltage-dependence of rate constants:
α(V ) = α0 e−(a1 +b1 V )/RT
β(V ) = β0 e−(a2 +b2 V )/RT ,
(17)
where a1 , a2 , b1 , and b2 are constants specific for this
transition. In the following, this form with simple exponential voltage dependence of rate constants will be
called linear thermodynamic model.
A further simplification is to consider that the conformational change consists of the movement of a freely
moving gating particle with charge q (Hodgkin and
Huxley, 1952; see also Borg-Graham, 1991). The forward and backward rate constants can be written as
α(V ) = A e−γ qF(V −VH )/RT
β(V ) = A e(1−γ )qF(V −VH )/RT ,
(18)
(14)
where the constant Ai corresponds to the free energy
that is independent of the electrical field, and the linear
term Bi V to the effect of the electrical field on isolated charges and rigid dipoles (Hill and Chen, 1972;
where γ is the relative position of the energy barrier
in the membrane (between 0 and 1), VH is the halfactivation voltage, and A is a fixed constant. This form
was introduced by Borg-Graham (1991) to parameterize Hodgkin-Huxley models. Its parameters are very
Nonlinear Thermodynamic Models of Ionic Currents
convenient for fitting experimental data: VH and q affect the steady-state activation and inactivation curves,
whereas A and γ affect only the time constant with no
effect on steady-state relations.
Linear Thermodynamic Model of the T-Current This
formalism was used to constrain the fitting of experimental data by using rate constants dictated by Eq. (17).
This procedure led to the following optimal expressions:
αm = 0.049 exp[444γm (V + 54.6)/RT ]
(19)
βm = 0.049 exp[−444(1 − γm )(V + 54.6)/RT ]
(20)
αh = 0.00148 exp[−559γh (V + 81.9)/RT ]
(21)
βm = 0.00148 exp[559(1 − γh )(V + 81.9)/RT ],
(22)
where γm = 0.90 and γh = 0.25. These expressions
are drawn in Fig. 2 (thin dashed lines).
This model provided a good fit of the steady-state
relations (Fig. 2A–B, thin dashed lines), but the fit to
time constants was poor (Fig. 2C–D, thin dashed lines).
265
In particular, it was not possible to capture the saturation of τm and τh to constant values for depolarized
membrane potentials. This poor fit had catastrophic
consequences, as illustrated in Fig. 4A. Due to the
near-zero time constants at depolarized levels, the current activated and inactivated too fast and led to peak
current amplitudes that were over an order of magnitude smaller than the Hodgkin-Huxley model with
same channel densities (compare Fig. 4A with Fig. 1B).
We conclude that linear thermodynamic models do not
provide an acceptable behavior in voltage-clamp for
the T-current.
Nonlinear Thermodynamic Models In general, the effect of the electrical field on a protein will depend on the
number and position of its charged amino acids, which
will result in both linear and nonlinear components in
the free energy. Without assumptions about the underlying molecular structure, the free energy of a given
state i can be written as a Taylor series expansion of
the form
G i (V ) = Ai + Bi V + Ci V 2 + . . . ,
(23)
where Ai corresponds to the free energy that is independent of the electrical field, and the linear term Bi V
Figure 4. Voltage-clamp behavior of thermodynamic models of the T-current. Left panels: Activation protocol. Right panels: Protocol for the
recovery from inactivation (same protocol as in Fig. 1). A: Linear thermodynamic model of the T-current. B: Nonlinear thermodynamic model.
The same density of T-channels was used in all cases and was the same as the empirical models shown in Fig. 1B ( P̄Ca = 3 × 10−6 cm/s).
266
Destexhe and Huguenard
corresponds to the interaction between electrical field
with isolated charges and rigid dipoles, as described
above. The higher-order terms describe effects such as
electronic polarization and pressure induced by V (Hill
and Chen, 1972; Stevens, 1978; Andersen and Koeppe,
1992), as well as mechanical constraints (see below).
In general, each conformational state of the ion channel protein will be associated with a given distribution of charges and will therefore be characterized by a
given set of coefficients in Eq. (23). This is also true for
the activated state, which is a particular case of conformation. Applying Eqs. (12) and (23), the rate constant
can be written more generally as
r (V ) = r0 e−[(A
= r0 e
∗
+B ∗ V +C ∗ V 2 +...)−(A0 +B0 V +C0 V 2 +...)]/RT
−(a+bV +cV 2 +...)/RT
,
,
(24)
where a = A∗ − A0 , b = B ∗ − B0 , c = C ∗ − C0 , . . . ,
represent differences between the linear and nonlinear
components of the free energy of the initial and activated states (according to Eq. (23)).
Applying these considerations to the reversible
open-closed transition (Eq. (16)) leads to the following
general expression for the voltage-dependence of rate
constants:
α(V ) = α0 e−(a1 +b1 V +c1 V
2
+...)/RT
β(V ) = β0 e−(a2 +b2 V +c2 V
2
+...)/RT
,
(25)
where a1 , a2 , b1 , b2 , c1 , c2 , . . . , are constants specific
of this transition.
It is important to note that, in general, these parameters are not necessary interrelated because the three
different conformations implicated here (initial, activated, final as in Fig. 3) may have very different distributions of charges, resulting in different coefficients
in Eq. (23) and thus also resulting in different values
for a1 , . . . , c2 . In the following, this general functional
form for the voltage dependence of rate constants will
be called nonlinear thermodynamic model.
In the low field limit (during relatively small transmembrane voltages), the contribution of the higherorder terms may be negligible, and one can approximate the free energy in various ways. The simplest approximation would only consider first-order terms in
V , which gives the linear thermodynamic model (Eq.
(17)). Other approximations would consist in quadratic
or higher-order expansions of the free energy. These
approximations include nonlinear effects of the voltage on the free energy, and may lead to saturating rate
constants, as we will show below.
For example, the quadratic expansion of Eq. (25)
can be written as
α(V ) = A e−[b1 (V −VH ) + c1 (V −VH ) ]/RT
2
(26)
β(V ) = A e[b2 (V −VH ) + c2 (V −VH ) ]/RT ,
2
and similarly, its cubic expansion:
α(V ) = A e−[b1 (V −VH ) + c1 (V −VH )
β(V ) = A e[b2 (V −VH ) + c2 (V −VH )
2
2
+ d1 (V −VH )3 ]/RT
+ d2 (V −VH )3 ]/RT
,
(27)
where A, b1 , . . . , d2 are constants as defined above.
In addition to the effect of voltage on isolated charges
or rigid dipoles, described in Eq. (17), these forms
account for more sophisticated effects such as electronic polarization or the deformation of the protein
by the electrical field (Hill and Chen, 1972; Stevens,
1978). This contribution, however, was estimated to be
small (see Sigworth, 1993). Higher-order terms would
also take into account the presence of mechanical constraints on the gating process, as illustrated in Fig. 5.
If gating results from the movement of a freely moving
charge (Fig. 5A), the free energy will depend linearly
on voltage. However, if gating depends on the movement of a gating charge subject to a mechanical constraint (modeled here as a spring of constant k; Fig. 5B),
the force due to this constraint will contribute for a nonlinear term in the free energy (−kV 2 in the case of a
spring). This model is probably more realistic than using a freely moving charge because charged residues in
general are strongly constrained by the structure of the
protein and are therefore not moving freely. Adding
more complex constraints will likely results in adding
further nonlinear contributions to the free energy (such
as the third-order terms in Eq. (27)).
Nonlinear Thermodynamic Model of the T-Current
Nonlinear thermodynamic models (Eq. (25)) of different complexity were considered to fit the voltage-clamp
data of the T-current. The quadratic expansion still provided a poor fit of the time constants, although better
than linear fits (not shown). Acceptable fits were obtained for a cubic expansion of the rate constants, given
by
αm (V ) = Am exp − [bm1 (V − Vm ) + cm1 (V − Vm )2
+ dm1 (V − Vm )3 ]/RT
βm (V ) = Am exp − [bm2 (V − Vm ) + cm2 (V − Vm )2
+ dm2 (V − Vm )3 ]/RT
(28)
Nonlinear Thermodynamic Models of Ionic Currents
267
Figure 5. Models of ion-channel gating based on the movement of an electric charge inside the channel. A: A freely moving gating charge
will result in a free energy that depends linearly on voltage. B: Imposing onstraints on the movement of the gating charge will add nonlinear
terms in the free energy (see text). The example shown here illustrates the case of a gating charge attached to a spring of constant k, which will
result in a quadratic voltage-dependence of the free energy.
αh (V ) = Ah exp − [bh1 (V − Vh ) + ch1 (V − Vh )2
+ dh1 (V − Vh ) ]/RT
3
βh (V ) = Ah exp − [bh2 (V − Vh ) + ch2 (V − Vh )2
+ dh2 (V − Vh )3 ]/RT.
The best fit of this nonlinear thermodynamic model
is shown in Fig. 2 (thick dashed lines) and was obtained
with the following parameters: Am = 0.053 ms−1 , Vm =
−56 mV, bm1 = −260 J mV−1 , cm1 = 2.20 J mV−2 ,
dm1 = 0.0052 J mV−3 , bm2 = 64.85 J mV−1 , cm2 =
2.02 J mV−2 , dm2 = 0.036 J mV−3 , Ah = 0.0017 ms−1 ,
Vh = −80 mV, bh1 = 163 J mV−1 , ch1 = 4.96 J mV−2 ,
dh1 = 0.062 J mV−3 , bh2 = −438 J mV−1 , ch2 = 8.73
J mV−2 , dh2 = −0.057 J mV−3 . Figure 2 (thick dashed
lines) shows that this model could capture the form
of the voltage dependence of the time constants. In
particular, it could fit the saturating values for the time
constants at depolarized levels. Nonlinear expansions
of higher order provided better fits, but the difference
was not qualitative (not shown).
Using these rate constants to simulate the voltageclamp experiments on the T-current produced accept-
able behavior, as shown in Fig. 4B. All protocols of
activation (Fig. 4B1), deactivation (not shown), inactivation (not shown), and recovery from inactivation
(Fig. 4B2) showed a similar voltage dependent behavior as the experimental data.
Genesis of Low-Threshold Calcium Spikes Finally,
the different models shown above for the T-current
were compared in current clamp. A single compartment
model of the thalamic cell was generated (see Eq. (10)
in Section, Methods) and contained leak currents
and the T-current using one of the models described
above. The genesis of the low-threshold calcium spike
was monitored through return to rest after injecting
hyperpolarizing currents. The Hodgkin-Huxley type
model of the T-current generated low-threshold spikes
in a grossly all-or-none fashion (Fig. 6A). The linear thermodynamic model (Fig. 6B) did not generate
low-threshold spikes, consistent with the very small
amplitude of the current evidenced above (Fig. 4A). On
the other hand, the nonlinear thermodynamic model
of the T-current presented a behavior more consistent with the Hodgkin-Huxley type model (Fig. 6C).
268
Destexhe and Huguenard
Figure 6. Low-threshold spikes generated by three different models of the T-current. Comparison of the same current-clamp simulation for
three different Hodgkin-Huxley type models for the T-current: an empirical model (A), a linear thermodynamic model (B) and a nonlinear
thermodynamic model (C). The simulation consisted in injecting hyperpolarizing current pulses of various amplitudes (−0.025, −0.05, −0.075,
−0.1, −0.125, and −0.15 nA) and of 1 sec duration. At the end of the pulse, the model generated a low-threshold spike on return to rest. D:
Peak amplitude of low-threshold spikes (LTS) generated by the different models of the T-current. All simulations were done using the same
single-compartment geometry that constained leak currents in addition to the T-current (see Section 2, Methods). The density of T-channels was
identical in all cases ( P̄Ca = 5 × 10−5 cm/s) and was in the range of densities estimated from rat ventrobasal thalamic neurons (Destexhe et al.,
1998).
Although the shape of the low-threshold spikes were
not identical, their peak amplitudes were remarkably
similar (Fig. 6D). We therefore conclude that nonlinear thermodynamic models provide fits of comparable
quality to empirical Hodgkin-Huxley type models, using a functional form of the voltage-dependence derived from physically plausible arguments.
4.
Discussion
In this article, we have shown that (1) Hodgkin-Huxley
type models in which the transition rates are deduced
from a linear dependence of the free energy on voltage
fail to account for the voltage-dependent properties of
some currents, such as the T-current in thalamic neurons, and (2) models in which the free energy depends
nonlinearly on voltage can capture these properties. We
discuss below possible applications of these nonlinear
thermodynamic models.
4.1.
Applications to Hodgkin-Huxley Models
As we have shown for the T-type calcium current, an
important caveat of linear thermodynamic models is
that the time constants may be arbitrarily close to zero.
This is due to the fact that rate constants described by
simple exponentials of voltage (Eq. (17)) can reach arbitrarily large values.2 Different solutions to this problem have been proposed (Borg-Graham, 1991; Willms
et al., 1999) and consisted in artificial alterations of the
Hodgkin-Huxley formalism by either setting minimal
Nonlinear Thermodynamic Models of Ionic Currents
values for the rate constants or forcing the rate constants
to saturate at extreme voltages. However, these procedures are largely empirical, which contrasts with the
initial aim of thermodynamic models.
In reality, the rate of conformational changes of
proteins are necessarily bounded by mechanical constraints and consequently, the transition time constants
have a minimum value (e.g, Fig. 2C–D, symbols). We
have explored here the hypothesis that this minimum
value is due to nonlinear effects of the voltage on the
ion channel. We have examined functional forms for
the voltage dependence of rate constants that include
effects such as deformation of the protein by the electric
field (Hill and Chen, 1972; Stevens, 1978; Andersen
and Koeppe, 1992) or mechanical constraints on the
gating process (Fig. 5). These considerations lead to
a more general form of the voltage dependence of the
free energy. The resulting model can capture the saturation of the rate constants at some voltages and naturally
leads to time constants saturating to a minimal value.3
This type of model can therefore capture experimental
data showing such a minimal time constant, as we have
illustrated for the T-type current (Fig. 2C–D, symbols).
It must, however, be kept in mind that models in
which the free energy is represented by a quadratic or
third-order polynomial of V are approximations. In
its nonapproximated form (Eq. 23), the free energy
can be expressed as an infinite-order polynomial of V
(Stevens, 1978). Approximating this form by a secondor third-order polynomial relies on the hypothesis that
the voltage is not too large (the “low-field limit”). Consequently, these forms are adequate only for describing
the behavior of the channel in a specific range of membrane potential. The example shown here for the Tcurrent suggests that this range is large enough to apply
to the usual range of membrane potential used in macroscopic voltage-clamp experiments. Outside this range,
such as for very large voltages, the values of rate constants may diverge, and one must consider more complex expansions of the free energy, taking into account
more sophisticated effects of the voltage on the channel
structure.
Although nonlinear thermodynamic models capture
the kinetics and voltage dependence of currents with
an accuracy comparable to Hodgkin-Huxley models
based on empirical functions, they may not be correct because the Hodgkin-Huxley formalism itself may
be inadequate. It is possible that activation and invactivation are coupled, similarly to Na+ channels
(Armstrong, 1981; Aldrich et al., 1983; Bezanilla,
269
1985). In this case, the transitions leading to activation and inactivation should rather be modeled using
Markov kinetic representations, as was done for the Tcurrent in fibroblasts (Chen and Hess, 1990). Singlechannel recordings should be used to decide which
representation is most appropriate. In the case of the
thalamic T-type current, such data are not yet available.
At present, therefore, nonlinear thermodynamic models constitute an acceptable biophysical representation
for this current, which is consistent with macroscopic
voltage-clamp measurements in thalamic neurons.
4.2.
Applications to Markov Models
The nonlinear thermodynamic formalism is not specifically related to the Hodgkin-Huxley model but is also
applicable to any type of voltage-dependent model, including Markov representations. For example, when
macroscopic voltage-clamp data reveal that the time
constants saturate to a minimum value for activation,
it is usually assumed (e.g., Chen and Hess, 1990) that
this behavior requires two successive transitions:
α(V )
k1
β(V )
k2
−→
C1 −→
←− C2 ←− O,
(29)
where C1 and C2 are two distinct closed states of the
gate, and the rate constants α and β are simple exponentials of voltage (Eq. (17)). Because the second
transition is not dependent of voltage, it will act as a
rate-limiting factor when α and β reach high values. In
this case, the system will be governed essentially by k1
and k2 , which impose a limit on the rate of activation
and inactivation.
Although this scheme may be more realistic, it is
still unrealistic that the simple exponential representation for α and β permits the first transition to occur
arbitrarily fast at some voltages. This problem can be
solved by using nonlinear thermodynamic models for
α and β, leading to a simpler scheme with only one
transition:
α(V )
C1 −→
←− O.
(30)
β(V )
Here, the rate constants α and β are exponentials with
an argument depending nonlinearly on voltage (Eq.
(25)). Thus, in this case, the minimum value of the time
constant is not interpreted as due to the existence of a
270
Destexhe and Huguenard
rate-limiting transition but as due to mechanical constraints in the effect of the electrical field on the protein.
Nonlinear thermodynamic models therefore open the
possibility of describing the gating of ion channels using more compact models with few states, which may
be of great value for network simulations (Destexhe
et al., 1994).
Acknowledgments
Research supported by grants from the Medical Research Council of Canada and the National Institutes
of Health.
Notes
1. The term thermodynamics is used here to describe nonequilibrium
phenomena (e.g., de Groot and Mazur, 1962; Kreuzer, 1981),
which typically apply to ion channels during voltage-dependent
transitions.
2. Very fast transition rates are however common in single channels.
For a recent modeling of this aspect at the single-channel level,
see Sigg et al. (1999).
3. In addition, to be more realistic, models with a minimal time
constant are also more stable numerically.
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